X ELECTRIC FIELDS AND MATTER 1.1 Dielectics and dipoles It is an expeimentally obseved fact that if you put some (insulating) matte between the plates of a capacito then the capacitance inceases. Since the capacitance is given by the atio of chage to voltage C = Q/V this means that when matte is inseted between the plates o Q inceases when V is held constant V deceases when Q is held constant. The fist case would apply to a capacito connected to a battey, and the second case applies to an electically isolated capacito. We will conside the second. The explanation fo the effect is that thee is a elative displacement of the positive and negative chages in the substance between the plates. The chage on the positive plate attacts the negative chage and epels the positive chage of the substance, while the chage on the negative plate attacts the positive chage and epels +++++++++++++++ the negative chage of the substance. In this way thee is a patial cancellation of the capacito s electic field and the potential diffeence between the plates, and its capacitance theeby inceases. +++++++++++++++ The (insulating) mateial between the plates is called a dielectic. The enhancement of the capacitance of a capacito by a dielectic is thus a consequence of the polaisation of the dielectic. So to undestand the natue of dielectics, we must be familia with the popeties of polaised matte. The simplest model fo polaisation is a dipole. This is an equal positive and negative point chage sepaated by a small displacement. Thee ae two oigins fo dipoles in matte: 1) Pemanent dipoles, fo example NaCl Thee is a pemanent dipole due to the asymmetic chage distibution Na + Cl associated with the ionic bond of the NaCl molecule. The sodium ion loses an electon and the chloine ion gains it. While the chage distibution is asymmetic, the entie assembly emains electically neutal. ) Induced dipoles, fo example O (in an electic field) Hee an applied electic field induces a polaisation by pulling the positive + + E chages one way and the negative chages the othe way. This can happen in individual atoms and in covalently bonded molecules, as O, whee the bonding electon obits become distoted by the electic field. We shall see that dipoles can explain most of the electical popeties of insulatos. PH4 / B Cowan 1 1.1
X Electic Fields and Matte We define the electic dipole moment p as l p = q l (1.1) q p + q whee p points along l, fom q to +q. In the next section we will stat to exploe the popeties of electic dipoles. Howeve, hee we will conside the simple application of Gauss s law to a dipole. Gauss s law tells us that the flux of E though a closed suface is popotional to the total chage in the enclosed volume. Consideing a suface enclosing a dipole, we see that the total emegent flux is zeo. To zeoth ode the dipole has no electical effect: all the lines of E leaving the positive chage teminate on the negative one; thee ae no fee lines. But what is impotant is the fist non-vanishing ode. Although thee is no net flux, in some places the lines of E go in, and in othe places they go out as we shall see. 1. Electic potential of a dipole As a pelude to finding the electic field of a dipole we shall calculate the electic potential at the field point F by adding the potentials fom the two souce chages. field point F The sepaation of the chages is x, which we had denoted by l. The magnitude of the dipole moment in this case is then A a T p = xq. The potential at point F is given by q 1 1 V =. (1.) 4πε b a x x We ae consideing the potential at a distance fom the dipole S + q souce point -q vey much geate than the sepaation x between the chages. Afte all, in pactice is a macoscopic distance while x is a micoscopic dimension. In this case we see that the distances a and b ae vey simila. To the extent that they ae equal we see that thee is no potential, and thus no field. We must thus conside specifically the diffeence between (the ecipocals) of these distances. Howeve, since x/ is such a small quantity, we may conside an expansion in this small quantity, keeping only the leading tem. Expessions fo a and b may be witten in tems of the distance and the angle θ by the use of the cosine ule: a = + x + xcosθ = + x xcosθ so taking the squae oot we obtain a as B b b PH4 / B Cowan 1 1.
X Electic Fields and Matte and its ecipocal: x x a = 1+ cosθ + 1 1 x x = 1+ cosθ + a 1 1 We may now expand though the use of the binomial theoem: 1 1 x 1 x = 1 cos θ... a + with a simila expession fo b by changing the sign of the cosine: 1 1 x 1 x = 1 + cos θ +... b Fom these we find the diffeence in the ecipocal distances as 1 1 x = cos θ +... b a and so to leading ode in the small quantity x/ the electic potential at point F due to the dipole at point S is given by V q x = cos θ. 4πε But since the dipole moment p is simply xq, the potential may be expessed as V 1 p cosθ = (1.3) 4πε Note that fo the dipole V ~ 1/, compaed with V ~ 1/ fo a point chage (electic monopole). We shall see that this gives an E field that vaies as 1/ 3 fo the dipole compaed with 1/ fo the monopole. The dipole field thus falls off much slowe; it has a longe ange. Taking account of the diection of the dipole, since pcosθ = p.ˆ = p. we can wite the dipola potential in vecto fom as PH4 / B Cowan 1 1.3
X Electic Fields and Matte 1 p.ˆ V = 4πε (1.4) 1 p. =. 3 4πε It is inteesting to note fom ou definition and popeties of the gad function that since the change in a quantity V on undegoing a displacement d is given by Equation (3.4): dv = d. gad(v), the potential of the dipole may be witten in the compact fom: V 1 1 = p. gad. (1.5) 4πε 1.3 Electic field of a dipole E Since the electic field is a vecto, we must decide what coodinate system to use when descibing E in tems of its components. While E T ectangula coodinates ae the most familia, the symmety of this paticula system makes pola coodinates much moe convenient to use. Recall that the electic field is given as the negative gadient of the electostatic potential: E = gadv. p We ae familia with the expession fo the components of the gadient in ectangula coodinates; these wee given in Equation (3.3). You should have encounteed the gadient in othe coodinate systems fom you maths couses. In pola coodinates the components of gadv ae given by V 1 V E =, Eθ = (1.6) θ so that using V fo the dipole, fom Equation (1.3), we find E = pcosθ psinθ, E 4πε = 4 (1.7) 3 θ 3 πε PH4 / B Cowan 1 1.4
X Electic Fields and Matte The diagam shows the lines of E suounding an electic dipole. Obseve the chaacteistic shape of the dipole field. Although thee is much cancellation of the fields fom the two chages, it is not complete: + thus the dipole field. The almost-cancellation may be - clealy seen by supeimposing the fields fom the two chages. The lines fom one chage point out while those fom the othe point in. Whee the lines coss thee is almost complete cancellation, with the emnant field essentially pependicula. Thus the dipole field lines follow the Moié cuves of the intesections of the lines. + 1.4 Dipole in a unifom E field F E = q E p q + q T Fo a dipole p in a unifom E field we conclude: F = +q E In a unifom electic field the foces on the two chages constituting the dipole will be equal and opposite, although not co-linea. Thus thee will be no net foce on the dipole but thee will be a couple (a toque). Foce = (1.8) Couple = E q dl sinθ = p E. (1.9) We shall now calculate the potential enegy of a dipole in a unifom electic field. The potential enegy of the dipole is the wok done in binging it to its paticula oientation fom some efeence state. The toque on the dipole due to the extenal E field has magnitude pesinθ and it acts in the clockwise diection. So to otate the dipole we must apply PH4 / B Cowan 1 1.5
X Electic Fields and Matte an anticlockwise toque: in the diection of inceasing θ. And the wok done is the integal of the toque with espect to θ: Wok done = Toque dθ = Ep sin θ dθ = Ep cos θ. We have used the indefinite integal as we do not want to have the complication of consideing the initial efeence oientation; the above esult is coect to within the addition of a constant. It may be seen that the constant is zeo when the efeence state is taken as that when the dipole is pointing pependicula to the field. Then the potential enegy has the simple fom: V = E.p. (1.1) (Note that V hee is the potential enegy and not the electic potential.) 1.5 Dipole in a non-unifom E field In a unifom field thee was no net foce on a dipole because the foce on one chage was + q exactly balanced by the foce on the othe. This was a because both chages expeienced the same electic field. E F = + q E(a) If, howeve, the field wee non-unifom then the two p chages would be subject to diffeent foces esulting in T F = q E(b) a net foce on the dipole. We may wite (the x b component) of this foce as q F x = q { E x (a) E x (b) } = q { change in E x in going fom b to a }. But we know that the change in a scala field quantity, hee applied to a vecto component, upon displacement a distance dl is given by Equation (3.4): Thus we have fo the foce de x = dl. gad E x. F x = qdl. gad E x. Since qdl is the dipole moment p, and applying simila aguments fo the y and z diections, the components of the foce ae given as F x = p. gad E x, F y = p. gad E y, (1.11) F z = p. gad E z. These may be conveniently combined togethe in the following symbolic manne: F = (p. gad)e. (1.1) PH4 / B Cowan 1 1.6
X Electic Fields and Matte 1.6 Fee and bound chage We now come to a most impotant concept in the consideation of the electical popeties of matte. Ou fundamental viewpoint is that matte is simply a collection of chages and cuents. Howeve, the chages that constitute matte ae diffeent fom the chages that ae esponsible fo chaging an insulato o which flow as cuent in a conducto. We ae led to the idea of bound chage. This is the chage within neutal matte which, when displaced, is esponsible fo its polaisation: causing fo the dipole moments of the constituent atoms o molecules. It is convenient hee to intoduce the polaisation P of a macoscopic body. This is defined as the dipole moment pe unit volume. So if thee ae N dipoles pe unit volume, and each dipole has a moment p then the polaisation is given by P = Np. (1.13) 1.7 Bound suface chage density If an insulato has a polaisation then the displaced bound chage poduces a net positive chage at one end and a negative chage at the othe end. aea a P The dipole moment of the stip is is given by its volume times P: dipole moment = a l P. But this stip may be egaded as a single dipole: chages +q and q sepaated by a distance l. Thus lq = alp so that the equivalent chage on the ends of the stip ae +ap and ap. The aea of the stip is a so that the bound suface chage density σ b is simply P. Taking into account the oientation of the suface with espect to the diection of the polaisation the bound chage on the aea a may be witten as and the suface chage density as l Q b = P. a (1.14) σ = P.n. ˆ (1.15) b Hee ˆn is the unit vecto pointing pependiculaly fom the suface. 1.8 Bound volume chage density If the polaisation of a body is not unifom then thee will be a non-zeo chage density within the body even though as a whole it is electically neutal. When the body becomes polaised thee is an intenal movement of bound chage. Applying the aguments of the pevious PH4 / B Cowan 1 1.7
X Electic Fields and Matte section to a small element within the body, the bound chage flowing out of a volume v acoss a suface aea da is given by P.da. So the total chage flowing out of the volume is Q out = w P. d a. closed suface But the volume was neutal. So the chage emaining in the volume is then Q out. And this must be equal to the volume integal of the bound chage (volume) density ρ b : volume ρ dv = b w P. da. bounding suface The ight hand side of this equation may be tansfomed to a volume integal though the use of the divegence theoem P. da = divpdv so that w bounding suface volume ρ bdv = divp dv. volume volume Now the volume of integation is eally quite abitay. This equation holds iespective of the pecise volume specified, and this can only hold if the aguments of the two integals ae equal. In othe wods, the bound chage ρ b is elated to P, the polaisation, by ρ b = div P. (1.16) (Thee ae some analogies between this discussion and the deivation of the equation of continuity.) 1.9 Gauss s law including dielectics Ou statement of Gauss s law, which we may expess as div E = ρ/ε includes the effect of all chages. In ou teatment of matte we intepet the electical popeties of dielectics as due to the action of bound chages. In othe wods we ae saying that thee ae two types of chage we must conside: bound chages and fee chages. And Gauss s law above efes to all the chages. We thus wite it in the fom div E = ρ fee /ε + ρ b /ε. But ρ b is elated to P by Equation (1.16), so that div E = ρ fee /ε div P/ε. Now the bound chage, o equivalently the polaisation, is something that we do not have contol ove; it just happens. It can be egaded as an effect athe that a cause, and as such we can take the div P ove to the left hand side of the equation, witing PH4 / B Cowan 1 1.8
X Electic Fields and Matte div(ε E + P) = ρ fee. (1.17) This equation is giving us anothe way to look at Gausss s law in the pesence of dielectics. Wheeas in the absence of dielectics the (fee) electic chage is the souce of the E field, in the pesence of a dielectic of polaisation P the fee chage may be egaded as the souce of a field E + P/ε. Conventionally one defines a new vecto field D, called the electic displacement: D = ε E + P. (1.18) In tems of this, Gauss s law in the pesence of matte may be witten as div D = ρ fee. (1.19) This is the modification of the Maxwell equation (M1) to include the effects of matte. Note that only the Maxwell equations with souces will become modified to account fo the effects of the bound souce, be it chage o cuent. Thus the div E equation is modified but the cul E equation will not be. And similaly in the magnetic case the div B equation will not be modified, while the cul B equation will be. 1.1 Fields in matte The discussions which led to the intoduction of the electic displacement D ae valid in matte on the aveage. It is eally atificial to talk of the dipole moment pe unit volume, which was how we defined the polaisation vecto E + + + + P. In eality, on the micoscopic level, at most places within matte the polaisation is zeo, only ising to a lage value at the atomic sites. Similaly, the electic field in matte is a vey apidly vaying quantity. Since the discussion above only makes sense fo continuously vaying polaisation P, we ae eally efeing to a spatial aveage ove volumes containing a lage numbe of atoms. That means that when using the field equations in thei fom div D = ρ fee, cul E = B/ t we ae efeing to spatially aveaged fields; these ae two of the macoscopic Maxwell equations. Howeve, when we use the field equations in thei oiginal fom div E = ρ/ε, cul E = B/ t then we mean the actual fields which can vay as apidly as necessay; these ae two of the micoscopic Maxwell equations. 1.11 Dielectic constant and electic susceptibility Fom the phenomenological point of view the macoscopic Maxwell equations have equied the use of a new electic field vecto, the D field. But this pesents a complication, because PH4 / B Cowan 1 1.9
X Electic Fields and Matte now thee ae moe vaiables than thee ae equations. We need anothe elationship connecting D and E. In geneal we will have some functional fom D = D(E) (1.) which is efeed to as a constitutive elation. This is a statement about the dielectic medium in which we measue the fields. Fo a linea, isotopic homogeneous dielectic the E field and the D field will be popotional and paallel. In that case we can wite D = εe (1.1) whee ε is known as the pemittivity of the dielectic. Altenatively, this can be expessed as D = ε ε E (1.) whee ε is the pemittivity of fee space and ε is called the elative pemittivity o dielectic constant. Clealy ε is dimensionless. Fo ai, ε is about 1.6 at NTP, while fo wate it is about 81 fo static fields. In pactice the dielectic constant is found to vay with fequency fo oscillatoy fields; the polaisation of the atoms can t keep up as the E field vaies. The emaining discussions of this section efe specifically to LIH systems. We stat with a consideation of what is happening at the micoscopic level. Fo the dipole moment of an atom o molecule we wite p = αe (1.3) whee α is efeed to as the polaisibility. This is a elation that holds at the micoscopic level. At the macoscopic level the polaisation P will then be popotional to E and we wite P = ε χ e E, (1.4) and χ e is known as the electic susceptibility. The inclusion of ε in the definition means that χ will be dimensionless. The elation between the elative pemittivity and the electic susceptibility is easily seen to be ε = 1 + χ e. (1.5) 1.1 Fields at boundaies An impotant poblem, when consideing dielectics, is to undestand how the electic fields E and D behave at the bounday between two mateials. Apat fom the inteest in the static case, whee one might need to know the field in the vicinity of a suface, this poblem is elated to the efaction of electomagnetic waves at the bounday between diffeent media. It tuns out that the bounday conditions on E and D can be found easily fom the elevant Maxwell equations. In the static case and in the absence of fee chages, the two electic field equations ae div D =, cul E =. PH4 / B Cowan 1 1.1
X Electic Fields and Matte Applying div D = to the Gaussian pillbox, the total flux of D though the suface is zeo. If we let the height of the box shink to zeo then the flux of D in though the bottom D must be equal to the flux of D out though the top. And since the flux n of D though the suface is given by the nomal component of D times the aea and the aeas of the top and the bottom ae equal, this means that the nomal component of D on both sides of the bounday must be equal. In othe wods the nomal component of D is continuous at a bounday. Obviously, the bounday can be between two dielectics o between one dielectic and fee space. Applying cul E = to the loop indicated, we may evaluate the line integal of E aound the loop. As the height of the loop tends to zeo only the tangential pats of the E integal contibute. The integal on one side of the bounday cancels t with the one on the othe side. And since the diections of integation ae opposite, it follows that the two line integals (evaluated in the same diection) must be equal. The lengths ae equal and so theefoe the components of E along the suface must be equal. In othe wods the tangential component of E at a bounday is continuous. 1.13 Coulomb s law evisited In a vacuum the foce between two chages is given by Coulomb s law. We now investigate how this will be changed in the pesence of a dielectic. Clealy we ae consideing the case whee the two chages ae immesed in a fluid (liquid o gas) dielectic. In a solid thee would be othe stesses involved when embedded chages wee moved even slightly. In the vacuum case we went fom the invese squae law fo E: Q E = 4πε to the Gauss s law expession giving the divegence of E: div E = ρ/ε. Now in the pesence of a dielectic medium Gauss s law, in geneal, becomes athe complicated since the bound chages of the medium must be included in ρ. Howeve, we have seen that the difficulty may be side-stepped by the intoduction of the electic displacement vecto D. We then ecove a Gauss s law expession, fo D, in tems of the fee chages div D = ρ fee. Fo isotopic media, whee the lines of the field will emanate fom a chage unifomly in all diections, the expession fo div D may be inveted, to give Q D = 4π whee the chage Q hee is the fee chage, and note that thee is no ε in the expession. PH4 / B Cowan 1 1.11
X Electic Fields and Matte The foce on a second chage q in the dielectic is given by F = qe, so that we have to find the E field coesponding to the value of D given above. We have aleady assumed the dielectic is isotopic, to obtain the expession fo D. Making the futhe assumptions of lineaity and homogeneity allows us to elate E to D though the pemittivity o dielectic constant: E = D/ε = D/ε ε so that the foce is given by qq F =. 4πεε This is identical to the oiginal Coulomb s law, except that now we have the absolute pemittivity ε = ε ε athe than the fee space value ε. Thus the foce is simply educed, in invese popotion to the elative pemittivity. Note, howeve, that the simplicity of the analogy is somewhat illusoy; the esult is only valid fo LIH dielectics. 1.14 Compaison of dielectics and fee-space Whee necessay we assume LIH mateials. Fee space 1) div E = ρ ε Dielectics div D = ρ div E = ρ fee εε fee Q ˆ ) = 4πε E E = 4πεε 3) F = qe F = qe qq ˆ 4) = 4πε F F = 4πεε Q qq ˆ ˆ E n = σ / ε E n = σ / ε ε 5) fee space conducto σ d ielectic conducto σ 6) V = ρ ε V = ρ εε 7) E n = σ / ε E n = σ / ε ε C = ε A/d C = ε ε A/d PH4 / B Cowan 1 1.1
X Electic Fields and Matte It is appopiate to make a numbe of obsevations about the above table of esults. The left hand column, efeing to fee space descibes the behaviou of the micoscopic field E in tems of the souces, which ae all the chages pesent. In contast to this, the ight hand column, efeing to dielectics, is descibing the macoscopic o spatially aveaged fields E and now D as well. The souces ae now the (spatially aveaged) fee chages alone. The bound chages ae accounted fo in the polaisation P, which contibutes to the D field. Of paticula impotance is the expession fo the electic foce F = qe. This is tue in both the micoscopic and the macoscopic pictue, although the intepetation is diffeent in the two cases. Fom the micoscopic point of view both E and F may be vey apidly vaying quantities (in the vicinity of the elementay chages which constitute matte). Howeve, fom the macoscopic point of view both the foce F and the electic field E ae egaded as spatial aveages and they have negligible vaiation on the atomic scale. When you have completed this chapte you should: undestand that a dielectic inceases the capacitance of a capacito appeciate that this is elated to polaisation of the dielectic; undestand the distinction between pemanent and induced electic dipoles; appeciate that most of the electical popeties of insulato can be undestood in tems of dipoles; undestand the definition of electic dipole moment; appeciate that the field of a dipole esults fom the almost cancellation of the fields of the positive and negative chages of the dipole; be able to calculate the electic potential of a dipole; be able to calculate the electic field of a dipole; undestand that in a unifom electic field a dipole expeiences a toque but no foce; undestand the expession fo the enegy of an electic dipole in an electic field; undestand that in a non-unifom electic field the dipole will expeience a foce and be able to calculate this foce; appeciate the distinction between fee and bound chage; PH4 / B Cowan 1 1.13
X Electic Fields and Matte undestand the definition of polaisation of a macoscopic body; undestand that a polaised body has an equivalent bound suface chage density; undestand that in a body with non-unifom polaisation, thee is an equivalent bound volume chage density; be able to calculate the bound volume chage density in tems of the divegence of the polaisation; appeciate how Gauss s law may be e-expessed in the pesence of dielectics and the electic displacement; be familia with and undestand the definition of the D field; undestand the distinction between micoscopic and macoscopic electic fields; undestand the distinction between the micoscopic and the macoscopic field equations; know of the definitions of dielectic constant and electic susceptibility; undestand the concept of a constitutive elation; know the meaning of polaisibility; undestand and be able to pefom calculations on the change in E and D at boundaies between dielectics; know how Coulomb s law is modified in the pesence of (LIH) dielectics; appeciate the vaious modifications of the popeties of fee space in the pesence of dielectics. PH4 / B Cowan 1 1.14